The difference between the “good” and “bad” choice of the
operator \(\Omega \)  mapping \({\cal H}_{math}\)  onto \({\cal L}\)  is strongly model-dependent. One of the conditions of success is
that this mapping (often called Dyson mapping) has to be as general
and adaptable as possible, i.e., in particular, non-unitary
(cf. (REF )).
In a way explained in [1]} the new, preconditioned,
user-friendlier Hamiltonian \(H\)  can be perceived as defined in
another Hilbert space \({\cal H}_{math}\)  which is, due to property
(REF ), not equivalent to its physical predecessor \({\cal L}\) : In the Dyson's realistic calculations [2]}, for
example, the fermionic states in \({\cal L}\)  were represented by the
bosonic states in \({\cal H}_{math}\) .
